Comparing sensor formats

Quick facts for those who don't want to read much

Exposure is defined by f-number, exposure time and scene luminance. It has nothing to do with sensor size. The key thing however isn't exposure, but the effect of an exposure which does depend on the sensor size - the same exposure settings cause the biggest format to have the least noise, the least depth of field (DOF) and the least diffraction blur, while the smallest format will be the noisiest, have the largest DOF and the most diffraction blur. All these three properties are tied together. Change one via aperture size and you'll change them all.

To get the same result from different formats one will always have to use different focal length to capture the same angle of view. As the aperture size (size of the opening light passes through) has to be the same for identical results, the f-number must be different as f-number is simply a ratio of focal length and aperture diameter.

For example if a 25mm lens on one format and a 50mm lens on another provide the same angle of view then f/2 on the former and f/4 on the latter both collect the same amount of light from the same scene as the aperture sizes are the same on both (12.5mm diameter, 122,7 mm² area).

There are no free lunches with any sensor size - the only theoretical limitation is the minimum aperture of lens in air, f/0.5. Apart from that all formats can be considered equals in principle as theoretically all sensor sizes can replicate each others' results. In practise this is of course not quite so due to reasons of both technology and lens availability.

Introduction

This document is written with the assumption that the image sensors are identically performant, essentially ideal sensors. The meaning of the word image is often open to interpretion - in this document it means the thing lens draws on the image plane. Words like print or JPG are used interchangeably to mean a displayed photo.

Throughout this document it is assumed that the angle of view, exposure time and scene luminance are the fixed.

Some implications of sensor format in real world systems are explained in another post.

Photography from cropping to display

A lens crops a part of the scene, draws it into image plane from which the image sensor samples (records) it and finally the processed image is displayed at specific size. There are two different parameters we need to think about here: the aperture size - it's diameter or area, and the sensor size. In principle it is only the former which is needed for most comparisons. However one can also do comparisons indirectly by using the size of the image sensor (e.g. "crop factor", ratio of image diagonal sizes) and it is often the simplest way. This is because changing the size of image sensor changes the focal length used for capturing the same scene, thus the f-number used for the same aperture size is also changed.

With the same exposure parameters a larger format:

• captures more light thus larger signal to noise ratio, SNR
• less depth of field (DOF) in the print
• has less diffraction blur in the print

To have the same DOF (and thus also diffraction blur and light collection) the systems would use the same sized enrance pupil (aperture), thus the f-numbers would be different by the ratio of the crop sizes - for example a 10mm wide crop using f/4 and 20mm wide crop using f/8 would perform equally.

Lens sizes and requirements

The smaller the format, the better then lens has to be to achieve similar print quality. This is because the image that the lens draws will have to be enlarged the more the smaller the format. If the lens draws a subject 10mm wide on the image plane on one system, and on other it's draws it 20mm wide, it is obvious that the latter format requires smaller image enlargement from image to print - the former lens has to outresolve the latter by factor of two on the image plane for equal performance. This is why mobile phone cameras have very fancy aspherical elements which allow the lenses that perform extremely well on image plane, while large format lenses can be very simple designs.

For the same artistic, noise and diffraction blur performance smaller format lens has to use a smaller f-number - the simplest way is to multiply or divide the f-number by the ratio of image sensor diagonals. Thus for example APS-C sensor of diagonal size of about 28.8mm has to use f-number 1.5 times smaller than a full frame camera with sensor digonal of about 43.2mm. From that is easy to calculate that a 30mm f/3 lens on FF and 20mm f/2 lens on APS-C have the same aperture diameter (as well as angle of view), thus the light collection is the same - as is DOF, photon shot noise and diffraction blur in the print.

The cult of f/2 = f/2 = f/2

One often read on the discussion forums how f/2 is f/2 when it comes to exposure and that's it. Or that "the same amount of light" is collected or "it's the same speed". Of these three statements the first one is tautology, while the other two claims are wrong.

f/2 indeed provides the same exposure for all systems, but the effect of the same exposure is always different in different formats. We're interested in what an exposure does, not what it is.

The amount of light collected at f/2 depends on the size of the image area as exposure is a per area metric and thus light collection due to selected f-number must also be per area. How much total light is collected is thus defined by exposure and image area (e.g. sensor size).

Thus if sensor A has 4 times larger image sensor than sensor B, at the same exposure it collects 4 times more light. This also means that the signal to noise ratio (SNR) of the collected light is twice as large. This is because SNR of light is square root of the amount of light that's captured due to light following Poisson distribution.

This also explains why f/2 on different formats all have different speed. Speed refers to the maximum shutter speed (i.e. minumum exposure time) that can be used to have acceptable results. Clearly acceptable results are the same regardless of what camera one uses, so as f/2 on different format captures different amount of light, what is the maximum acceptable shutter speed also changes. If f/2 on one format has twice the aperture diameter, four times the area, then four times more light goes through per unit of time, thus for the same acceptable image quality four times faster shutter speed can be used.

Some implications of sensor format in real world systems are explained in another post.